There are two preconditions for solving equations by Cramer's law. One is that the number of equations should be equal to the number of unknowns, and the other is that the determinant of the coefficient matrix is not equal to zero.

Solving equations by Cramer's law is actually equivalent to solving linear equations by inverse matrix method, and the relationship between the solutions of linear equations and their coefficients and constants is established. However, it is often very heavy to calculate the n-order n+ 1 determinant when solving, so Cramer's rule is mostly used for theoretical proof and rarely used for concrete solution.

2. Matrix elimination method

The augmented matrix of linear equations is transformed into a simplified trapezoidal matrix by elementary transformation of rows. Then the linear equations with simplified step matrix as augmented matrix have the same solution as the original equations. When the equations have a solution, the unknown quantity corresponding to the unit column vector is regarded as a non-free unknown quantity, and the rest of the unknown quantity is regarded as a free unknown quantity, so that the solution of the linear equations can be obtained.

Extended data

Xj stands for unknown quantity, aij stands for coefficient, and bi stands for constant term.

Called coefficient matrix and augmented matrix. If x 1=c 1, x2=c2, …, xn=cn are substituted into a given equation, then (c 1, c2, …, cn) is called a solution. If c 1, c2, …, cn are not all 0, then (c 1, c2, …, cn) is called a non-zero solution.

If the constant terms are all 0, it is called homogeneous linear equations, and it always has zero solution (0, 0, …, 0). Two equations, if they have the same number of unknowns and the same solution set, are called homosolution equations. The main problems discussed in linear equations are:

When will the equations have a solution?

The number of solutions of equations with solutions.

By solving the equation, the structure of the solution is determined. These problems have been solved satisfactorily: if the given equations have solutions, then rank (A)= rank (augmented matrix); If rank (A)= rank = r, then there is a unique solution when r=n; When r < n, there are infinite solutions; It can be solved by elimination.

When the nonhomogeneous linear equations have a solution, the only necessary and sufficient condition of the solution is that the corresponding homogeneous linear equations have only zero solution; The necessary and sufficient condition of infinite solution is that homogeneous linear equations have non-zero solutions. On the other hand, when the derived non-homogeneous linear equations have only zero and non-zero solutions, the original equations may not have unique or infinite solutions. In fact, the equation may not have a solution at this time.

Clem's rule (see determinant) gives a formula for solving a special system of linear equations. The solution set of any homogeneous equation with n unknowns constitutes a subspace of n-dimensional space.

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